Chapter 7: Conic Sections and Their Geometry (Set-3)
Circle through points A(0,0)A(0,0), B(2,0)B(2,0), C(0,2)C(0,2) has equation
A x2+y2−2x−2y=0x2+y2−2x−2y=0
B x2+y2+2x+2y=0x2+y2+2x+2y=0
C x2+y2−4x−4y=0x2+y2−4x−4y=0
D x2+y2−2=0x2+y2−2=0
A circle through origin has form x2+y2+2gx+2fy=0x2+y2+2gx+2fy=0. Using (2,0)(2,0) gives 4+4g=0⇒g=−14+4g=0⇒g=−1. Using (0,2)(0,2) gives 4+4f=0⇒f=−14+4f=0⇒f=−1.
Line x+y=2x+y=2 is tangent to circle x2+y2=2×2+y2=2 because
A Center lies on line
B Line passes origin
C Distance equals radius
D Two intersections exist
Center is (0,0)(0,0) and radius is 22. Distance from origin to line x+y−2=0x+y−2=0 is 22=222=2. Equal distance and radius means tangency.
For circle x2+y2+4x−6y+9=0x2+y2+4x−6y+9=0, radius is
A 11
B 22
C 33
D 55
Here g=2,f=−3,c=9g=2,f=−3,c=9. Radius squared =g2+f2−c=4+9−9=4=g2+f2−c=4+9−9=4? Wait: careful—complete square: (x+2)2+(y−3)2=4(x+2)2+(y−3)2=4. So radius =2=2.
Completing squares gives (x+2)2+(y−3)2=4(x+2)2+(y−3)2=4. Hence center (−2,3)(−2,3) and radius r=4=2r=4=2. This avoids sign mistakes from direct formula use.
Chord of contact from P(x1,y1)P(x1,y1) to x2+y2=r2x2+y2=r2 is
A xx1+yy1=0xx1+yy1=0
B x+x1=rx+x1=r
C xx1+yy1=r2xx1+yy1=r2
D y+y1=ry+y1=r
For a circle centered at origin, the chord of contact from external point P(x1,y1)P(x1,y1) is T=0T=0: xx1+yy1=r2xx1+yy1=r2. It joins the two tangency points of tangents from PP.
If two circles intersect orthogonally, then for their radii r1,r2r1,r2 and centers distance dd
A d2=r12+r22d2=r12+r22
B d=r1+r2d=r1+r2
C d2=r12−r22d2=r12−r22
D d=r1−r2d=r1−r2
For orthogonal intersection, the radii to an intersection point are perpendicular, making a right triangle with sides r1,r2r1,r2 and hypotenuse dd between centers. So d2=r12+r22d2=r12+r22.
Radical axis of circles S1=0S1=0, S2=0S2=0 is given by
A S1+S2=0S1+S2=0
B S1S2=0S1S2=0
C S1−S2=0S1−S2=0
D S12+S22=0S12+S22=0
Points on radical axis have equal power w.r.t both circles. Power difference is S1−S2S1−S2. Setting it to zero gives a linear equation, hence radical axis is a straight line.
Tangent length from P(5,0)P(5,0) to x2+y2=9×2+y2=9 is
A 3434
B 55
C 22
D 44
Center is origin, radius r=3r=3, and OP=5OP=5. Tangent length =OP2−r2=25−9=16=4=OP2−r2=25−9=16=4. This applies only when point is outside circle.
A circle touches xx-axis if its center (h,k)(h,k) and radius rr satisfy
A ∣k∣=r∣k∣=r
B ∣h∣=r∣h∣=r
C h=kh=k
D hk=rhk=r
Distance from center to xx-axis is ∣k∣∣k∣. Touching means distance equals radius. Therefore ∣k∣=r∣k∣=r. Similarly, touching yy-axis would require ∣h∣=r∣h∣=r.
For circle (x−1)2+(y+2)2=25(x−1)2+(y+2)2=25, tangent at point (6,−2)(6,−2) is
A y=−2y=−2
B x−1=0x−1=0
C x=6x=6
D y+2=0y+2=0
Center is (1,−2)(1,−2). Point (6,−2)(6,−2) lies horizontally from center, so radius there is horizontal. Tangent is perpendicular to radius, hence vertical line through point: x=6x=6.
If a line cuts a circle in two points, the power of point on line (outside segment) is
A Negative
B Zero
C Undefined
D Positive
Power is PT2PT2 for tangency and equals product of directed intercepts for secant. For a point outside the circle, OP>rOP>r, so OP2−r2>0OP2−r2>0, giving positive power.
Parabola with focus (0,3)(0,3) and directrix y=−3y=−3 is
A x2=6yx2=6y
B x2=12yx2=12y
C y2=12xy2=12x
D y2=6xy2=6x
Vertex is midway between focus and directrix, at origin. Here a=3a=3. Standard form for upward parabola is x2=4ayx2=4ay. So x2=12yx2=12y.
For parabola y2=8xy2=8x, value of aa is
A 11
B 44
C 22
D 88
Compare y2=8xy2=8x with y2=4axy2=4ax. Then 4a=8⇒a=24a=8⇒a=2. Hence focus is (2,0)(2,0), directrix is x=−2x=−2, and latus rectum length is 88.
Tangent to y2=8xy2=8x at point (2,4)(2,4) is
A y=x+2y=x+2
B 2y=x+62y=x+6
C y=2xy=2x
D y=xy=x
Here a=2a=2. Tangent at (x1,y1)(x1,y1) is yy1=2a(x+x1)yy1=2a(x+x1). So 4y=4(x+2)⇒y=x+24y=4(x+2)⇒y=x+2.
Using yy1=2a(x+x1)yy1=2a(x+x1) with a=2,(x1,y1)=(2,4)a=2,(x1,y1)=(2,4): 4y=4(x+2)⇒y=x+24y=4(x+2)⇒y=x+2. It touches parabola at (2,4)(2,4) and has slope 1.
Normal to y2=4axy2=4ax at parameter tt passes through point
A (−at2,2at)(−at2,2at)
B (at2,2at)(at2,2at)
C (2at2,−at)(2at2,−at)
D (0,at)(0,at)
Normal at parameter tt for y2=4axy2=4ax is y=−tx+2at+at3y=−tx+2at+at3. Setting y=0y=0 gives x=2at2x=2at2. Then yy value on normal at that xx is −at−at, so it passes through (2at2,−at)(2at2,−at).
Pair of points on parabola y2=4axy2=4ax that form latus rectum are
A (2a,±a)(2a,±a)
B (a,±2a)(a,±2a)
C (a,±a)(a,±a)
D (0,±2a)(0,±2a)
Latus rectum is chord through focus (a,0)(a,0) perpendicular to axis. Put x=ax=a in y2=4axy2=4ax gives y2=4a2y2=4a2, so y=±2ay=±2a. Endpoints are (a,±2a)(a,±2a).
Equation of parabola with vertex (1,2)(1,2) opening upward is of type
A (y−2)2=4a(x−1)(y−2)2=4a(x−1)
B (x−2)2=4a(y−1)(x−2)2=4a(y−1)
C (y−1)2=4a(x−2)(y−1)2=4a(x−2)
D (x−1)2=4a(y−2)(x−1)2=4a(y−2)
Upward opening means axis parallel to yy-axis and squared term in xx. Translation from origin to (1,2)(1,2) gives (x−1)2=4a(y−2)(x−1)2=4a(y−2). Sign of aa decides up/down.
For parabola x2=4ayx2=4ay, length of focal chord at parameter tt equals
A 2a(1+t2)2a(1+t2)
B 4a(1−t2)4a(1−t2)
C 4a(1+t2)4a(1+t2)
D 2a(1−t2)2a(1−t2)
For x2=4ayx2=4ay, focal chord corresponding to parameter tt has endpoints at parameters tt and −1/t−1/t. Its length simplifies to 4a(1+t2)4a(1+t2), a standard medium-level parabola result.
If a parabola has directrix x=1x=1 and focus (5,0)(5,0), its vertex is
A (3,0)(3,0)
B (2,0)(2,0)
C (4,0)(4,0)
D (6,0)(6,0)
Vertex lies midway between focus and directrix along the axis direction. Here both are along xx-axis. Midpoint between x=5x=5 and line x=1x=1 is x=3x=3, so vertex is (3,0)(3,0).
The focal distance from any point on parabola to focus equals distance to
A Center
B Vertex
C Directrix
D Axis
Parabola is defined by focus-directrix property: distance from a point to focus equals perpendicular distance to directrix. This definition ensures eccentricity e=1e=1 and is valid for every point on the parabola.
For y2=4axy2=4ax, chord joining parameters t1,t2t1,t2 has equation
A y(t1−t2)=2(x−at1t2)y(t1−t2)=2(x−at1t2)
B y(t1+t2)=2(x+at1t2)y(t1+t2)=2(x+at1t2)
C y=tx+aty=tx+at
D y2=4axy2=4ax
Using parametric points (at2,2at)(at2,2at), the chord joining t1,t2t1,t2 has standard form y(t1+t2)=2(x+at1t2)y(t1+t2)=2(x+at1t2). It reduces to tangent when t1=t2t1=t2.
For ellipse x225+y29=125×2+9y2=1, eccentricity is
A 4/54/5
B 3/53/5
C 5/45/4
D 2/52/5
Here a2=25⇒a=5a2=25⇒a=5, b2=9⇒b=3b2=9⇒b=3. Then c2=a2−b2=16⇒c=4c2=a2−b2=16⇒c=4. Eccentricity e=c/a=4/5e=c/a=4/5.
For ellipse x2a2+y2b2=1a2x2+b2y2=1, director circle equation is
A x2+y2=a2+b2x2+y2=a2+b2
B x2+y2=2a2x2+y2=2a2
C x2+y2=a2−2b2x2+y2=a2−2b2
D x2+y2=b2−a2x2+y2=b2−a2
For ellipse, points from which tangents are perpendicular lie on director circle given by x2+y2=a2−2b2x2+y2=a2−2b2. It is real only if a2>2b2a2>2b2; otherwise no real director circle exists.
Length of latus rectum of ellipse x2a2+y2b2=1a2x2+b2y2=1 is
A 2a2/b2a2/b
B 2b2/a2b2/a
C b2/ab2/a
D 2ab2ab
Latus rectum is focal chord perpendicular to major axis. For standard ellipse, its full length is 2b2a2ab2. It becomes smaller as aa increases compared to bb, reflecting increased “stretch.”
For ellipse, equation of normal at (x1,y1)(x1,y1) involves
A Only x1x1
B Only y1y1
C Only aa
D Both coordinates
Normal depends on slope of tangent, which depends on x1,y1x1,y1. For ellipse, tangent slope is −b2x1a2y1−a2y1b2x1 (when y1≠0y1=0), so normal uses both coordinates.
If ellipse major axis is along yy-axis, its standard form is
A x2b2+y2a2=1b2x2+a2y2=1
B x2a2+y2b2=1a2x2+b2y2=1
C x2a2−y2b2=1a2x2−b2y2=1
D y2b2−x2a2=1b2y2−a2x2=1
Major axis aligns with the variable having larger semi-axis aa. If major axis is along yy, then y2y2 term must have a2a2 in denominator. Hence x2b2+y2a2=1b2x2+a2y2=1.
For ellipse, distance from center to directrix equals
A aeae
B b/eb/e
C a/ea/e
D bebe
For an ellipse with eccentricity e=c/ae=c/a, directrices are x=±a/ex=±a/e (major along xx-axis). Thus distance from center to directrix is a/ea/e. This comes from focus-directrix definition.
For ellipse, product of focal distances at endpoints of minor axis equals
A b2b2
B a2a2
C c2c2
D abab
At minor axis endpoint (0,b)(0,b), distances to foci (±c,0)(±c,0) are equal: c2+b2=a2=ac2+b2=a2=a. Product becomes a⋅a=a2a⋅a=a2. This is a neat property.
Conjugate diameters in an ellipse correspond to directions of
A Parallel tangents
B Parallel normals
C Equal radii
D Equal foci
If two diameters are conjugate, then tangents at endpoints of one diameter are parallel to the other diameter’s direction. This connection between diameter direction and tangent direction is central to ellipse conjugate diameter geometry.
Area of ellipse using major axis length 2a2a and minor axis 2b2b is
A 2πab2πab
B π(a+b)π(a+b)
C πabπab
D πa2πa2
Semi-axes are aa and bb. Ellipse area is πabπab. It can be seen as scaling a unit circle area by factors aa and bb in perpendicular directions, multiplying area by abab.
Auxiliary circle helps relate ellipse point by
A Same radius always
B Same focus location
C Same directrix line
D Same angle parameter
Using auxiliary circle x2+y2=a2x2+y2=a2, a point on ellipse is obtained by dropping a perpendicular from circle point (acost,asint)(acost,asint) to ellipse giving (acost,bsint)(acost,bsint). Same parameter tt is used.
For hyperbola x216−y29=116×2−9y2=1, asymptotes are
A y=±43xy=±34x
B y=±916xy=±169x
C y=±34xy=±43x
D y=±169xy=±916x
For x2a2−y2b2=1a2x2−b2y2=1, asymptotes are y=±baxy=±abx. Here a=4,b=3a=4,b=3, so y=±34xy=±43x.
For hyperbola, equation of conjugate axis endpoints (standard) are
A (0,±b)(0,±b)
B (±b,0)(±b,0)
C (±c,0)(±c,0)
D (0,±c)(0,±c)
For x2a2−y2b2=1a2x2−b2y2=1, conjugate axis lies along yy-axis with semi-length bb. So its endpoints are (0,±b)(0,±b). These are not on the hyperbola, but define its rectangle.
For x2a2−y2b2=1a2x2−b2y2=1, tangent at (x1,y1)(x1,y1) meets xx-axis at
A x1/a2x1/a2
B b2/y1b2/y1
C a2/x1a2/x1
D y1/b2y1/b2
Tangent is xx1a2−yy1b2=1a2xx1−b2yy1=1. Put y=0y=0 to get xx1a2=1⇒x=a2x1a2xx1=1⇒x=x1a2. This is a useful intercept property.
Eccentricity of hyperbola x2a2−y2b2=1a2x2−b2y2=1 is
A 1−b2/a21−b2/a2
B b/ab/a
C a/ba/b
D 1+b2/a21+b2/a2
For hyperbola, c2=a2+b2c2=a2+b2, so e=c/a=1+b2/a2e=c/a=1+b2/a2. This is always greater than 1, matching hyperbola definition via focus-directrix ratio.
Rectangular hyperbola in standard axes form is
A x2+y2=a2x2+y2=a2
B x2−y2=a2x2−y2=a2
C x2=4ayx2=4ay
D x2a2+y2b2=1a2x2+b2y2=1
A rectangular hyperbola has perpendicular asymptotes, which happens when a=ba=b in x2a2−y2b2=1a2x2−b2y2=1, giving x2−y2=a2x2−y2=a2. Its asymptotes are y=±xy=±x.
Parametric form using hyperbolic functions for x2a2−y2b2=1a2x2−b2y2=1 is
A (asinhu,bcoshu)(asinhu,bcoshu)
B (acosu,bsinu)(acosu,bsinu)
C (acoshu,bsinhu)(acoshu,bsinhu)
D (asect,btant)(asect,btant)
Identity cosh2u−sinh2u=1cosh2u−sinh2u=1 matches hyperbola form. Taking x=acoshux=acoshu, y=bsinhuy=bsinhu satisfies x2a2−y2b2=1a2x2−b2y2=1. This gives all points on one branch.
If hyperbola has directrix x=a/ex=a/e, then focus is at
A x=aex=ae
B x=a/ex=a/e
C x=ax=a
D x=c/ex=c/e
For standard hyperbola along xx-axis, directrices are x=±a/ex=±a/e and foci are x=±aex=±ae. This comes from e=c/ae=c/a and the focus-directrix definition for conics.
Director circle of rectangular hyperbola x2−y2=a2x2−y2=a2 is
A x2+y2=a2x2+y2=a2
B x2+y2=0x2+y2=0
C x2+y2=2a2x2+y2=2a2
D x2−y2=2a2x2−y2=2a2
For hyperbola, director circle is x2+y2=a2+b2x2+y2=a2+b2. In rectangular case a=ba=b, so it becomes x2+y2=2a2x2+y2=2a2. It represents points from which tangents are perpendicular.
Conjugate hyperbola shares with given hyperbola the same
A Foci
B Asymptotes
C Directrices
D Vertices
Conjugate hyperbola x2a2−y2b2=−1a2x2−b2y2=−1 has the same asymptotes as the original because setting equation to zero gives the same lines. However, vertices and branches orientation differ.
For hyperbola, if a point lies between branches region, its power relative to hyperbola is
A Always zero
B Always positive
C Always negative
D Not standard
“Power of a point” is a standard concept for circles, not generally defined the same way for hyperbola. For other conics, related ideas use pole-polar or chord of contact, but “power” isn’t the common invariant.
Equation x2+y2−z2=0x2+y2−z2=0 represents a
A Sphere surface
B Cylinder surface
C Right circular cone
D Plane surface
x2+y2=z2x2+y2=z2 is a double cone with vertex at origin and axis along zz-axis. Cross-sections at constant zz are circles of radius ∣z∣∣z∣. It is homogeneous quadratic, confirming a cone.
Semi-vertical angle αα of cone x2+y2=z2tan2αx2+y2=z2tan2α satisfies
A tanα=r/∣z∣tanα=r/∣z∣
B sinα=r+zsinα=r+z
C cosα=r/z2cosα=r/z2
D tanα=z/rtanα=z/r
At height zz, circular radius r=x2+y2r=x2+y2. The generator makes angle αα with axis, so tanα=r∣z∣tanα=∣z∣r. Squaring gives the cone equation form used in 3D coordinate geometry.
A quadratic surface is a cone if its equation is
A Degree 1 homogeneous
B Degree 2 homogeneous
C Degree 3 homogeneous
D Only constant terms
A cone with vertex at origin must satisfy scaling property: if (x,y,z)(x,y,z) lies on it, so does (kx,ky,kz)(kx,ky,kz). This happens when the equation is homogeneous of degree 2, i.e., all terms total degree 2.
For Ax2+By2+Cz2+2Fyz+2Gzx+2Hxy=0Ax2+By2+Cz2+2Fyz+2Gzx+2Hxy=0, cone classification depends mainly on
A Linear terms
B Constant term
C Coefficients matrix
D Trig terms
This quadratic form can be represented by a symmetric matrix. Nature of cone (real/imaginary, pair of planes, etc.) depends on eigenvalues and sign pattern of this matrix. Linear/constant terms are absent in pure cone through origin.
Pair of planes occurs when a cone equation factors into
A Two linear factors
B Three linear factors
C Quadratic squared
D Circle equation
If a homogeneous quadratic in x,y,zx,y,z factors as (l1)(l2)=0(l1)(l2)=0, it represents two planes through origin. This is a degenerate cone. Checking factorization is a standard medium-level idea in 3D analytic geometry.
For general conic Ax2+Bxy+Cy2+⋯=0Ax2+Bxy+Cy2+⋯=0, rotation removes xyxy term when
A tanθ=BA+Ctanθ=A+CB
B sin2θ=ABsin2θ=BA
C cos2θ=BAcos2θ=AB
D tan2θ=BA−Ctan2θ=A−CB
Choosing angle θθ such that tan2θ=BA−Ctan2θ=A−CB makes the mixed term coefficient zero in rotated coordinates. This is a standard method to simplify conics by aligning axes with principal directions.
Condition for pair of straight lines in Ax2+2Hxy+By2=0Ax2+2Hxy+By2=0 is
A H2
B H2>ABH2>AB
C H2=ABH2=AB
D A=BA=B
The homogeneous quadratic represents pair of lines through origin if it factors. For Ax2+2Hxy+By2=0Ax2+2Hxy+By2=0, factorization is possible when discriminant H2−AB=0H2−AB=0. Then lines may be real and distinct or coincident.
If a conic has equation S=0S=0, then chord of contact from point PP is obtained by
A T=0T=0
B S=0S=0 only
C S2=0S2=0
D S+T=0S+T=0
In conic theory, T=0T=0 represents the polar (chord of contact) of point P(x1,y1)P(x1,y1) w.r.t S=0S=0. For circle it matches xx1+yy1=r2xx1+yy1=r2. This is the base of pole-polar ideas.
The polar of the center of a circle is
A Diameter line
B Tangent line
C Line at infinity
D Any secant
For a circle, the center lies inside and does not have real tangents. Its polar becomes the line at infinity in projective sense. In basic coordinate geometry, it is treated as “no real chord of contact,” hence infinity concept.
Two conics can intersect in maximum number of points equal to
A 22
B 44
C 33
D 66
Two second-degree curves lead to solving two quadratic equations simultaneously. In general, the system can have up to 4 intersection points (counting multiplicity). Special cases include tangency, overlap, or fewer real intersections.